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Simultaneous tridiagonalization of two symmetric matrices
Author(s) -
Garvey Seamus D.,
Tisseur Françoise,
Friswell Michael I.,
Penny John E. T.,
Prells Uwe
Publication year - 2003
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.733
Subject(s) - tridiagonal matrix , symmetric matrix , mathematics , matrix (chemical analysis) , identity (music) , rank (graph theory) , identity matrix , combinatorics , symmetry (geometry) , eigenvalues and eigenvectors , chemistry , physics , geometry , chromatography , quantum mechanics , acoustics
Abstract We show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non‐singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non‐orthogonal rank‐one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non‐orthogonal rank‐one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O ( n 3 ) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.